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Basic AC Power THREE-PHASE CIRCUITS THREE-PHASE CIRCUITS The design of three-phase AC circuits lends itself to a more efficient method of producing and utilizing an AC voltage. EO 1.5 STATE the reasons that three-phase power systems are used in the industry. EO 1.6 Given values for current, voltage, and power factor in a three-phase system, CALCULATE the following: a. Real power b. Reactive power c. Apparent power EO 1.7 Given a diagram of a wye- or delta-connected three- phase system, DESCRIBE the voltage/current relationships of the circuit. EO 1.8 STATE the indications of an unbalanced load in a three- phase power system. Three-Phase Systems A three-phase (3φ) system is a combination of three single-phase systems. In a 3φ balanced system, power comes from a 3φ AC generator that produces three separate and equal voltages, each of which is 120° out of phase with the other voltages (Figure 10). Rev. 0 Page 17 ES-09 THREE-PHASE CIRCUITS Basic AC Power Figure 10 Three-Phase AC Three-phase equipment (motors, transformers, etc.) weighs less than single-phase equipment of the same power rating. They have a wide range of voltages and can be used for single-phase loads. Three-phase equipment is smaller in size, weighs less, and is more efficient than single-phase equipment. Three-phase systems can be connected in two different ways. If the three common ends of each phase are connected at a common point and the other three ends are connected to a 3φ line, it is called a wye, or Y-, connection (Figure 11). If the three phases are connected in series to form a closed loop, it is called a delta, or ∆-, connection. ES-09 Page 18 Rev. 0 Basic AC Power THREE-PHASE CIRCUITS Figure 11 3φ AC Power Connections Power in Balanced 3φ Loads Balanced loads, in a 3φ system, have identical impedance in each secondary winding (Figure 12). The impedance of each winding in a delta load is shown as Z∆ (Figure 12a), and the impedence in a wye load is shown as Zy (Figure 12b). For either the delta or wye connection, the lines A, B, and C supply a 3φ system of voltages. Figure 12 3φ Balanced Loads Rev. 0 Page 19 ES-09 THREE-PHASE CIRCUITS Basic AC Power In a balanced delta load, the line voltage (VL) is equal to the phase voltage (Vφ ), and the line current (IL) is equal to the square root of three times the phase current ( 3 Iφ ). Equation (9-5) is a mathematical representation of VL in a balanced delta load. Equation (9-6) is a mathematical representation of IL in a balanced delta load. VL = Vφ (9-5) IL 3 Iφ (9-6) In a balanced wye load, the line voltage (VL) is equal to the square root of three times phase voltage ( 3 Vφ ), and line current (IL) is equal to the phase current (Iφ ). Equation (9-7) is a mathematical representation of VL in a balanced wye load. Equation (9-8) is a mathematical representation of IL in a balanced wye load. VL 3 Vφ (9-7) IL Iφ (9-8) Because the impedance of each phase of a balanced delta or wye load has equal current, phase power is one third of the total power. Equation (9-10) is the mathematical representation for phase power (Pφ ) in a balanced delta or wye load. Pφ = Vφ Iφ cosθ (9-10) Total power (PT) is equal to three times the single-phase power. Equation (9-11) is the mathematical representation for total power in a balanced delta or wye load. PT = 3Vφ Iφ cosθ (9-11) 3 IL In a delta-connected load, VL = Vφ and Iφ so: 3 PT 3 VL IL cos θ 3 VL In a wye-connected load, IL = Iφ and Vφ so: 3 PT 3 VL IL cos θ ES-09 Page 20 Rev. 0 Basic AC Power THREE-PHASE CIRCUITS As you can see, the total power formulas for delta- and wye-connected loads are identical. Total apparent power (ST) in volt-amperes and total reactive power (QT) in volt-amperes-reactive are related to total real power (PT) in watts (Figure 13). A balanced three-phase load has the real, apparent, and reactive powers given by: PT 3 VT IL cos θ Figure 13 3φ Power Triangle ST 3 VT IL QT 3 VT IL sin θ Example 1: Each phase of a delta- connected 3φ AC generator supplies a full load current of 200 A at 440 volts with a 0.6 lagging power factor, as shown in Figure 14. Find: 1. VL 2. IL 3. PT 4. QT 5. ST Figure 14 Three-Phase Delta Generator Solution: 1. VL Vφ VL 440 volts Rev. 0 Page 21 ES-09 THREE-PHASE CIRCUITS Basic AC Power 2. IL 3 Iφ (1.73)(200) IL 346 amps 3. PT 3 VL IL cos θ (1.73)(440)(346)(0.6) PT 158.2 kW 4. QT 3 VL IL sin θ (1.73)(440)(346)(0.8) QT 210.7 kVAR 5. ST 3 VL IL (1.73)(440)(346) ST 263.4 kVA Example 2: Each phase of a wye- connected 3φ AC generator supplies a 100 A current at a phase voltage of 240V and a power factor of 0.9 lagging, as shown in Figure 15. Find: 1. VL 2. PT 3. QT 4. ST Figure 15 Three-Phase Wye Generator ES-09 Page 22 Rev. 0 Basic AC Power THREE-PHASE CIRCUITS Solution: 1. VL 3 Vφ (1.73)(240) VL 415.2 volts 2. PT 3 VL IL cos θ (1.73)(415.2)(100)(0.9) PT 64.6 kW 3. QT 3 VL IL sin θ (1.73)(415.2)(100)(0.436) QT 31.3 kVAR 4. ST 3 VL IL (1.73)(415.2)(100) ST 71.8 kVA Unbalanced 3φ Loads An important property of a three-phase balanced system is that the phasor sum of the three line or phase voltages is zero, and the phasor sum of the three line or phase currents is zero. When the three load impedances are not equal to one another, the phasor sums and the neutral current (In) are not zero, and the load is, therefore, unbalanced. The imbalance occurs when an open or short circuit appears at the load. If a three-phase system has an unbalanced load and an unbalanced power source, the methods of fixing the system are complex. Therefore, we will only consider an unbalanced load with a balanced power source. Example: A 3φ balanced system, as shown in Figure 16a, contains a wye load. The line-to- line voltage is 240V, and the resistance is 40 Ω in each branch. Rev. 0 Page 23 ES-09 THREE-PHASE CIRCUITS Basic AC Power Figure 16 3φ Unbalanced Load Find line current and neutral current for the following load conditions. 1. balanced load 2. open circuit phase A (Figure 16b) 3. short circuit in phase A (Figure 16c) Vφ VL 1. IL Iφ Iφ Vφ Rφ 3 V L IL 3 Rφ 240 1.73 40 138.7 40 IL 3.5 amps IN 0 ES-09 Page 24 Rev. 0 Basic AC Power THREE-PHASE CIRCUITS 2. Current flow in lines B and C becomes the resultant of the loads in B and C connected in series. VL IB IC IB RB RC 240 40 40 IB 3 amps IC 3 amps IN IB IC 3 3 IN 6 amps VL 3. IB IC IB RB 240 40 IB 6 amps IC 6 amps The current in Phase A is equal to the neutral line current, IA = IN. Therefore, IN is the phasor sum of IB and IC. IN 3 IB (1.73)(6) IN 10.4 amps In a fault condition, the neutral connection in a wye-connected load will carry more current than the phase under a balanced load. Unbalanced three-phase circuits are indicated by abnormally high currents in one or more of the phases. This may cause damage to equipment if the imbalance is allowed to continue. Rev. 0 Page 25 ES-09 THREE-PHASE CIRCUITS Basic AC Power Summary Three-phase circuits are summarized below. Three-Phase Circuits Summary Three-phase power systems are used in the industry because: - Three-phase circuits weigh less than single-phase circuits of the same power rating. - They have a wide range of voltages and can be used for single-phase loads. - Three-phase equipment is smaller in size, weighs less, and is more efficient than single-phase equipment. Unbalanced three-phase circuits are indicated by abnormally high currents in one or more of the phases. ES-09 Page 26 Rev. 0

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